HOMEWORK 2 (Math 250 A, B)1.(a) Given dcbaA, show that: (20 pts) (i) 2)(xxpATr(A)x+ det(A) (ii) The eigenvalues of Aare 24)()(22,1bcdada.(b) Let Abe an nxn invertible matrix. Then, show that:(i) All the eigenvalues of Aare non-zero. (ii) The eigenvalues of 1Aare of the form 1, where is an eigenvalue of A.(iii) )/1()det()()(1xpAxxpAnA.(Hint: For (b) part (iii) , use the fact that det(AB) = det(A)det(B) )2. (a) If Ais an nxn diagonalizable matrix, then show that PDPAkk1, where k, D is diagonal, and Pis some invertible matrix . (20 pts) (b) Use part (a) to find 11A, where 2150010171A. (Hint: For (a), use the fact that since Ais diagonalizable then Ais similarto a diagonal matrix D) 3. Show that Aand Bare orthogonal vectors in an inner product space Vif and only if 222BAXwhereBAX. The above, is usually called the

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